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1 Package zic September 4, 2015 Version 0.9 Date Title Bayesian Inference for Zero-Inflated Count Models Author Markus Jochmann Maintainer Markus Jochmann Description Provides MCMC algorithms for the analysis of zero-inflated count models. The case of stochastic search variable selection (SVS) is also considered. All MCMC samplers are coded in C++ for improved efficiency. A data set considering the demand for health care is provided. License GPL (>= 2) Depends R (>= 3.0.2) Imports Rcpp (>= ), coda (>= ) LinkingTo Rcpp, RcppArmadillo NeedsCompilation yes Repository CRAN Date/Publication :07:45 R topics documented: docvisits zic zic.svs Index 8 1

2 2 docvisits docvisits Demand for Health Care Data Description Usage Format This data set gives the number of doctor visits in the last three months for a sample of German male individuals in The data set is taken from Riphahn et al. (2003) and is a subsample of the German Socioeconomic Panel (SOEP). In contrast to Riphahn et al. (2003) only male individuals from the last wave are considered. See Jochmann (2013) for further details. data(docvisits) This data frame contains 1812 observations on the following 22 variables: docvisits number of doctor visits in last 3 months age age agesq age squared / 1000 age30 1 if age >= 30 age35 1 if age >= 35 age40 1 if age >= 40 age45 1 if age >= 45 age50 1 if age >= 50 age55 1 if age >= 55 age60 1 if age >= 60 health health satisfaction, 0 (low) - 10 (high) handicap 1 if handicapped, 0 otherwise hdegree degree of handicap in percentage points married 1 if married, 0 otherwise schooling years of schooling hhincome household monthly net income, in German marks / 1000 children 1 if children under 16 in the household, 0 otherwise self 1 if self employed, 0 otherwise civil 1 if civil servant, 0 otherwise bluec 1 if blue collar employee, 0 otherwise employed 1 if employed, 0 otherwise public 1 if public health insurance, 0 otherwise addon 1 if add-on insurance, 0 otherwise

3 zic 3 References Jochmann, M. (2013). What Belongs Where? Variable Selection for Zero-Inflated Count Models with an Application to the Demand for Health Care, Computational Statistics, 28, Riphahn, R. T., Wambach, A., Million, A. (2003). Incentive Effects in the Demand for Health Care: A Bivariate Panel Count Data Estimation, Journal of Applied Econometrics, 18, Wagner, G. G., Frick, J. R., Schupp, J. (2007). The German Socio-Economic Panel Study (SOEP) Scope, Evolution and Enhancements, Schmollers Jahrbuch, 127, zic Bayesian Inference for Zero-Inflated Count Models Description zic fits zero-inflated count models via Markov chain Monte Carlo methods. Usage zic(formula, data, a0, b0, c0, d0, e0, f0, n.burnin, n.mcmc, n.thin, tune = 1.0, scale = TRUE) Arguments formula data A symbolic description of the model to be fit specifying the response variable and covariates. A data frame in which to interpret the variables in formula. a0 The prior variance of α. b0 The prior variance of β j. c0 The prior variance of γ. d0 The prior variance of δ j. e0 The shape parameter for the inverse gamma prior on σ 2. f0 The inverse scale parameter the inverse gamma prior on σ 2. n.burnin n.mcmc n.thin tune scale Number of burn-in iterations of the sampler. Number of iterations of the sampler. Thinning interval. Tuning parameter of Metropolis-Hastings step. If true, all covariates (except binary variables) are rescaled by dividing by their respective standard errors.

4 4 zic Details The considered zero-inflated count model is given by yi Poisson[exp(ηi )], ηi = α + x iβ + ε i, ε i N(0, σ 2 ), d i = γ + x iδ + ν i, ν i N(0, 1), y i = 1(d i > 0)yi, where y i and x i are observed. The assumed prior distributions are α N(0, a 0 ), β k N(0, b 0 ), k = 1,..., K, γ N(0, c 0 ), δ k N(0, d 0 ), k = 1,..., K, σ 2 Inv-Gamma (e 0, f 0 ). The sampling algorithm described in Jochmann (2013) is used. Value A list containing the following elements: alpha Posterior draws of α (coda mcmc object). beta Posterior draws of β (coda mcmc object). gamma delta sigma2 acc References Posterior draws of γ (coda mcmc object). Posterior draws of δ (coda mcmc object). Posterior draws of σ 2 (coda mcmc object). Acceptance rate of the Metropolis-Hastings step. Jochmann, M. (2013). What Belongs Where? Variable Selection for Zero-Inflated Count Models with an Application to the Demand for Health Care, Computational Statistics, 28, Examples ## Not run: data( docvisits ) mdl <- docvisits ~ age + agesq + health + handicap + hdegree + married + schooling + hhincome + children + self + civil + bluec + employed + public + addon post <- zic( f, docvisits, 10.0, 10.0, 10.0, 10.0, 1.0, 1.0, 1000, 10000, 10, 1.0, TRUE ) ## End(Not run)

5 zic.svs 5 zic.svs SVS for Zero-Inflated Count Models Description zic.svs applies SVS to zero-inflated count models Usage zic.svs(formula, data, a0, g0.beta, h0.beta, nu0.beta, r0.beta, s0.beta, e0, f0, c0, g0.delta, h0.delta, nu0.delta, r0.delta, s0.delta, n.burnin, n.mcmc, n.thin, tune = 1.0, scale = TRUE) Arguments formula data A symbolic description of the model to be fit specifying the response variable and covariates. A data frame in which to interpret the variables in formula. a0 The prior variance of α. g0.beta The shape parameter for the inverse gamma prior on κ β k. h0.beta The inverse scale parameter for the inverse gamma prior on κ β k. nu0.beta Prior parameter for the spike of the hypervariances for the β k. r0.beta Prior parameter of ω β. s0.beta Prior parameter of ω β. e0 The shape parameter for the inverse gamma prior on σ 2. f0 The inverse scale parameter the inverse gamma prior on σ 2. c0 The prior variance of γ. g0.delta The shape parameter for the inverse gamma prior on κ δ k. h0.delta The inverse scale parameter for the inverse gamma prior on κ δ k. nu0.delta Prior parameter for the spike of the hypervariances for the δ k. r0.delta Prior parameter of ω δ. s0.delta Prior parameter of ω δ. n.burnin n.mcmc n.thin tune scale Number of burn-in iterations of the sampler. Number of iterations of the sampler. Thinning interval. Tuning parameter of Metropolis-Hastings step. If true, all covariates (except binary variables) are rescaled by dividing by their respective standard errors.

6 6 zic.svs Details Value The considered zero-inflated count model is given by y i Poisson[exp(η i )], η i = α + x iβ + ε i, ε i N(0, σ 2 ), d i = γ + x iδ + ν i, ν i N(0, 1), y i = 1(d i > 0)y i, where y i and x i are observed. The assumed prior distributions are α N(0, a 0 ), β k N(0, τ β k κβ k ),, k = 1,..., K, κ β j Inv-Gamma(gβ 0, hβ 0 ), τ β k (1 ωβ )δ ν β + ω β δ 1, 0 ω β Beta(r β 0, sβ 0 ), γ N(0, c 0 ), δ k N(0, τ δ k κ δ k), k = 1,..., K, κ δ k Inv-Gamma(g δ 0, h δ 0), τ δ k (1 ω δ )δ ν δ 0 + ω δ δ 1, ω δ Beta(r δ 0, s δ 0), σ 2 Inv-Gamma (e 0, f 0 ). The sampling algorithm described in Jochmann (2013) is used. A list containing the following elements: alpha beta gamma delta sigma2 Posterior draws of α (coda mcmc object). Posterior draws of β (coda mcmc object). Posterior draws of γ (coda mcmc object). Posterior draws of δ (coda mcmc object). Posterior draws of σ 2 (coda mcmc object). I.beta Posterior draws of indicator whether τ β j I.delta omega.beta omega.delta acc Posterior draws of indicator whether τ δ j Posterior draws of ω β (coda mcmc object). Posterior draws of ω δ (coda mcmc object). Acceptance rate of the Metropolis-Hastings step. is one (coda mcmc object). is one (coda mcmc object).

7 zic.svs 7 References Jochmann, M. (2013). What Belongs Where? Variable Selection for Zero-Inflated Count Models with an Application to the Demand for Health Care, Computational Statistics, 28, Examples ## Not run: data( docvisits ) mdl <- docvisits ~ age + agesq + health + handicap + hdegree + married + schooling + hhincome + children + self + civil + bluec + employed + public + addon post <- zic.ssvs( mdl, docvisits, 10.0, 5.0, 5.0, 1.0e-04, 2.0, 2.0, 1.0, 1.0, 10.0, 5.0, 5.0, 1.0e-04, 2.0, 2.0, 1000, 10000, 10, 1.0, TRUE ) ## End(Not run)

8 Index Topic datasets docvisits, 2 docvisits, 2 zic, 3 zic.svs, 5 8

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